Biogeography-based optimization (BBO) is a new population-based evolutionary algorithm and is based on an old theory of island biogeography that explains the geographical distribution of biological organisms. BBO was introduced in 2008 and then a lot of modifications were employed to enhance its performance. This paper proposes two modifications; firstly, modifying the probabilistic selection process of the migration and mutation stages to give a fairly randomized selection for all the features of the islands. Secondly, the clear duplication process after the mutation stage is sized to avoid any corruption on the suitability index variables. The obtained results through wide variety range of test functions with different dimensions and complexities proved that the BBO performance can be enhanced effectively
without using any complicated form of the immigration and emigration rates. This essential modification has to be considered as an initial step for any other modification.
2. Computer Science & Information Technology (CS & IT) 142
Island, in biogeography, is any area of suitable habitat (local environment occupied by an
organism [5]) surrounded by an expense of unsuitable habitat and is endowed with exceptionally
rich reservoirs of endemic, exclusive, strange and relict species [6].
Each island has its own features as simple biotas, varying combinations of biotic and abiotic
factors, and variability in isolation, shape, and size [7,9].
In BBO, the islands (or habitats) are the dependent variables, and the features availability on
those islands represents the independent variables.
The original BBO has a weakness in its migration and mutation stages that affects its
performance. The objective of this paper is to solve that weakness by focusing on the root
problems instead of using other complicated approaches, such as the complicated migration
models represented in [13,16].
This paper is organized as follows: Section 2 gives a brief overview about the theory of island
biogeography and Section 3 explains the algorithm of BBO. Section 4 shows how BBO
performance can be enhanced more by solving the root problems of the migration and mutation
stages, and then followed by a performance comparison between the original and the modified
BBO in Section 5. The conclusions and suggestions are presented inSection 6.
2. THE THEORY OF ISLAND BIOGEOGRAPHY
The equilibrium theory of island biogeography proposes that the number of inhabited species on
an island is based on the dynamic equilibrium between new immigrated species onto an island
and the extinct species out from that island [2,3,8].
Fig. 1 graphically represents the equilibrium model with linear immigration (or speciation) rate λ
and emigration (or extinction) rate µ, where they can be plotted as logistic, exponential or any
proper function [4,10,11].
Figure 1. Simplified equilibrium model of a biota of a single island
I and E are the maximum possible immigration and emigration rates, respectively. Ioccurs when
the island is empty of any species and thus it offers a maximum opportunity to the species on the
other islands for immigrating to settle on it; whereas the arrivals on that island increases, the
3. 143 Computer Science & Information Technology (CS & IT)
opportunity for settlement will decrease, which means
as λ decreases, the species density increases, and thus the predation, competition and parasitism
factors will increase too; and as
maximum value E when λ reaches its minimum value [
MacArthur and Wilson, in their early study
I=E, where at time t, the recipient island has
immigration and emigration rates at the present of
ܲ௦ሺݐሻ to ܲ௦ሺݐ ∆ݐሻ can be described as:
ܲ௦ሺݐ ∆ݐሻ ൌ ܲ௦ሺݐሻሺ1 െ ߣ௦
From Eq. 1, to have S at time ሺݐ
1. S species at time t, and no immigration or emigration took place during the interval
2. (S - 1) species at time t, and one species immigrated;
3. (S + 1) species at time t, and one speci
The probability of immigration and/or emigration should not be more than one, and thus
be set with small value. Now, as
ௗೞሺ௧ሻ
ௗ௧
≅ lim∆௧→
ೞ
ௗೞሺ௧ሻ
ௗ௧
≅ െሺߣ௦ ߤ
By considering the above three conditions, Eq. 2 can be specified more as:
ܲ௦
ሶ ሺݐሻ ൌ ቐ
െሺߣ௦ ߤ௦ሻܲ௦ ߤ௦ା
െሺߣ௦ ߤ௦ሻܲ௦ ߣ௦ି
െሺߣ௦ ߤ௦ሻܲ௦
If ܲ௦ሺݐሻ is known, thenܲ௦
ሶ ሺݐሻ can be obtained from Eq. 3, where the value of
can be approximated as:
ܲ௦ሺݐ
Eq. 4 is the final confirmed form that has
∆ݐሻ.
For finding ܲ௦ሺݐሻ, Dan Simon in [1] gives two methods; either by solving Eq. 3 numerically, or
applying the following theorem:
Theorem 1: The steady-state value for the probability of the number of each species is given by:
Where ݒ and ݒ can be computed from the following eqs.:
Computer Science & Information Technology (CS & IT)
opportunity for settlement will decrease, which means that the immigration rate decreases. Also,
decreases, the species density increases, and thus the predation, competition and parasitism
; and as a result, the emigration rate µ will increase and reaches its
reaches its minimum value [12].
MacArthur and Wilson, in their early study [2,3], proposed a simplified equilibrium model with
, the recipient island has S species with probability Ps(t), and ߣ௦
migration rates at the present of S species in that island.Then the variation from
can be described as:
௦∆ݐ െ ߤ௦∆ݐሻ ܲ௦ିଵሺݐሻߣ௦ିଵ∆ݐ ܲ௦ାଵሺݐሻߤ௦ାଵ∆ݐ
∆,)ݐ one of the following three conditions should hold:
, and no immigration or emigration took place during the interval
, and one species immigrated;
, and one species emigrated.
The probability of immigration and/or emigration should not be more than one, and thus
be set with small value. Now, as ∆ݐ approaches 0, the ratio ቀ
∆ೞ
∆௧
ቁ approachesܲሶ௦ሺݐሻ:
ೞሺ௧ା∆௧ሻିೞሺ௧ሻ
∆௧
ߤ௦ሻܲ௦ሺݐሻ ߣ௦ିଵܲ௦ିଵሺݐሻ ߤ௦ାଵܲ௦ାଵሺݐሻ
conditions, Eq. 2 can be specified more as:
ାଵܲ௦ାଵ, ܵ ൌ 0
ିଵܲ௦ିଵ ߤ௦ାଵܲ௦ାଵ, 1 ܵ ܵ௫ െ 1
ሻ ߣ௦ିଵܲ௦ିଵ, ܵ ൌ ܵ௫
can be obtained from Eq. 3, where the value of ܲ௦ሺݐ
ሺ ∆ݐሻ ≅ ܲ௦ሺݐሻ ܲ௦
ሶ ሺݐሻ∆ݐ
Eq. 4 is the final confirmed form that has to be used in the program of BBO for calculating
, Dan Simon in [1] gives two methods; either by solving Eq. 3 numerically, or
state value for the probability of the number of each species is given by:
can be computed from the following eqs.:
that the immigration rate decreases. Also,
decreases, the species density increases, and thus the predation, competition and parasitism
and reaches its
, proposed a simplified equilibrium model with
andߤ௦ are the
hen the variation from
(1)
, one of the following three conditions should hold:
, and no immigration or emigration took place during the interval ∆;ݐ
The probability of immigration and/or emigration should not be more than one, and thus ∆ݐ has to
(2)
(3)
ሺ ∆ݐሻ in Eq. 1
(4)
BBO for calculating ܲ௦ሺݐ
, Dan Simon in [1] gives two methods; either by solving Eq. 3 numerically, or
state value for the probability of the number of each species is given by:
(5)
4. Computer Science & Information Technology (CS & IT) 144
ݒ ൌ [ݒଵ, ݒଶ, … , ݒௌೌೣାଵ]் (6)
ݒ ൌ
ܵ௫!
ሺܵ௫ 1 െ ݅ሻ! ሺ݅ െ 1ሻ!
ሺ݅ ൌ 1, … , ܵ௫ 1ሻ (7)
Although the second alternative is easier and ܲ௦ሺݐሻ can be computed directly without any
iteration, this method is not preferable in the most programs, such as C/C++, MATLAB, Octave,
Maple, Python, etc, because it is valid only when ܵ௫ 170, otherwise ܵ௫! ൌ ∞, unless an
additional sub-algorithm is used to overcome this problem. In addition, this approach will
consume extra CPU time for dealing with long product operations.
The remaining terms for finding ܲ௦ሺݐ ∆ݐሻ areߤ௦ and ߣ௦, which can be calculated directly as:
ߤ௦ ൌ
ܧ
ܵ௫
ܵ (8)
ߣ௦ ൌ 1 െ ߤ௦ ൌ ܫ ൬1 െ
ܵ
ܵ௫
൰ (9)
3. BIOGEOGRAPHY-BASED OPTIMIZATION (BBO)
BBO translates the natural distribution of species into a general problem solution [1]. Each island
represents one solution, where the good problem solution means that the island has lots of good
biotic "living: diversity of prey, trees, shrubs, meadow, etc" and abiotic "non-living: distance of
isolation, wind, temperature, humidity, water, area, etc" factors, which attracts more species than
the other islands [4]. Each featureis called suitability index variable (SIV), which represents the
independent variable of such a problem in BBO. As these features changes, the island suitability
index (ISI) changes too; thus in BBO, ISIis the dependent variable [1,17].
A problem with n-independent variables and k-islands or individuals can be expressed as:
ܫܵܫ ൌ ݂ሺܸܵܫଵ, ܸܵܫଶ, … , ܸܵܫሻ ݅ ൌ 1,2, … , ݇ (10)
In the early stages of introducing BBO, Dan Simon proposed four different types of migration
process, these types can be sorted as [15]:
1. Partial Migration Based BBO “PMB-BBO”
2. Single Migration Based BBO “SMB-BBO”
3. Simplified Partial Migration Based BBO “SPMB-BBO”
4. Simplified Single Migration Based BBO “SSMB-BBO”
From preceding study [20], it is shown that SMB-BBO and SSMB-BBO give poor performance
but with lowest CPU time, while the performance comparison between PMB-BBO and SPMB-
BBO shows that PMB-BBO gives better performance as the complexity, side constrains and/or
dimensions of a given problem increases and as the number of islands decreases; and vice versa
for SPMB-BBO.
In general, SPMB-BBO could trap in a local or at least a near-global optima, especially when the
mutation stage that compensate the weakness of its migration algorithm is absent. This drawback
led us to select PMB-BBO as a final confirmed BBO model for applying the proposed essential
modification,which will be explained in the next section.
5. 145 Computer Science & Information Technology (CS & IT)
The algorithm of BBO consists of two main stages, migration and mutation.
3.1. Migration
Considering Fig. 1 and Eq. 10, if island i has lots of features, then lots of species will colonize it,
which means that ߣ௦ becomes low and ߤ௦ becomes high.
Thus, the high ISI for island irepresents a good solution, and vice versa for a poor solution which
has a shortage in its features diversity, and reflected on the total available number of species;
where at this condition, ߣ௦ is high and ߤ௦ is low.
From Fig. 1, S1 is located beforeܵመ, where ߣ௦ is high, ߤ௦ is low and the solution ISI1 is poor; while
S2 is located after ܵመ, where ߣ௦ is low, ߤ௦ is high and the solution ISI2 is good. Based on that, ߣ௦
and ߤ௦ can be used as indications of poor and good solutions, respectively.
The purpose of migration process is to use high ISI islands as a source of modification to share
their features with low ISI islands, so the poor solutions can be probabilistically enhanced and
may become better than those good solutions.
The migration process of PMB-BBO can be described as:
Let ISIi denote the ith population member and contains n features
For each island ISIi (where i=1,2,3,…,k)
For each SIV s (where s=1,2,3,…,n)
Use ߣ to probabilistically select the immigrating island ISIi
If rand < ߣ
For j=1 to k
Use ߤ to probabilistically decide whether to emigrate to ISIi
If ISIj is selected
Randomly select an SIV σ from ISIj
Replace a random SIV s in ISIi with SIV σ
end if
end for
end if
nextSIV
next island
3.2. Mutation
In island theory, the species at equilibrium point ܵመ can be deviated dramatically due to some
external events. Events such as predators from other islands, tsunamis, volcanos, diseases or
earthquakes cause negative deviation, and the total number of species will steeply decreases [11].
On the other hand, there are some other useful events such as wind-carrying seeds or flotsams
6. Computer Science & Information Technology (CS & IT)
which provide good features
enhancement [12].In BBO, the mutation process is modeled as
count probabilities Ps , the mutation rate
݉
mmax is a user-defined maximum mutation rate that
From the previous equation, m reaches to its minimum "
vice versa. Thus, m is inversely proportional to
Fig. 2, where the species count S
solutions be mutated increases too.
Figure 2. Comparison between
During the mutation stage, the low and high
enhanced more than what they already have, where the solutions at the equilibrium point are not
mutated [1]. Even if the mutated solutions become worse, the optional sta
store the best solutions from one generation
The mutation process can be described as:
For ݅ ൌ 1 to k (where k is the number of islands, see Eq. 10)
Calculate probability Ps based on
Calculate mutation rate m (using Eq. 11)
Select ISIi with probability proportional to
If ISIi is selected
Replace SIV of ISIi with a randomly generated
end if
end for
Computer Science & Information Technology (CS & IT)
to an island, thus giving better solution with a significant
].In BBO, the mutation process is modeled as SIV mutation; and through species
, the mutation rate m can be determined as:
݉ ൌ ݉௫ ൬1 െ
ܲ௦
ܲ௫
൰
defined maximum mutation rate that m can reach, and Pmax = max (Ps).
reaches to its minimum "zero" at the maximum value of
is inversely proportional to Ps. This process can be graphically described
S starts from zero to Smax. As mmax increases, the chance to let the
solutions be mutated increases too.
. Comparison between Ps(t) and m(t) at different mmax
mutation stage, the low and high ISI solutions are likely to mutate, and then could be
enhanced more than what they already have, where the solutions at the equilibrium point are not
ven if the mutated solutions become worse, the optional stage, called elitism, will
store the best solutions from one generation to the next [17].
The mutation process can be described as:
(where k is the number of islands, see Eq. 10)
based on ߣ௦and ߤ௦ (by numerical or direct method)
(using Eq. 11)
probability proportional to Ps
with a randomly generated SIV
146
better solution with a significant
mutation; and through species
(11)
.
" at the maximum value of Ps, and
described as in
increases, the chance to let the
solutions are likely to mutate, and then could be
enhanced more than what they already have, where the solutions at the equilibrium point are not
ge, called elitism, will
7. 147 Computer Science & Information Technology (CS & IT)
3.3. BBO Algorithm
The BBO algorithm can be summarized through the flowchart of Fig.3.
The algorithm’s looping can be terminated either if reaches to an acceptable tolerance or after
completing the desired number of generations.
Figure 3. General flowchart of BBO algorithm
8. Computer Science & Information Technology (CS & IT) 148
4. THE PROPOSED ESSENTIAL MODIFICATION ON BBO
This paper tries to solve two fundamental problems that are associated with the original version or
the raw form of BBO with using just a linear immigration and emigration rates.
By accomplishing this modification, it can be used a basis for any further modification or
hybridization. This study shows that the linear immigration and emigration rates can do well if
the associated root problems are solved.
According to the original BBO program that was designed by Dan Simon in [21], there are two
essential parts that need to be solved:
4.1. Probabilistic Selection Process of the Migration and Mutation Stages
Referring to the preceding algorithms, the process for selecting SIVs of an island i that needs to be
migrated is done probabilistically, and the general code for this task is:
ܸܵܫ௦ሺ1: ݇ሻ ൌ 1[݀݊ݑݎ ሺ݊ െ 1ሻ × ݀݊ܽݎሺ1, ݊ሻ] (12)
If the independent variables of each ISI are represented as a vector of [ܸܵܫଵ, ܸܵܫଶ, … , ܸܵܫ], then
the analysis of the above code shows that the SIVs at the beginning and at the end have less
weight than the other (n-2) SIV that are located in between.
Fig. 4a shows how unfairly selection be done for each SIV of island i. In this example; n=5 and
k=1000, with 4 trails.
In MATLAB, the proposed modification is to use integer random“randi” instead of using
rounded real random “rand” with n SIV alignment, as in Eq. 12.This integer random function
provides pseudorandom integers from a uniform discrete distribution on 1to n.
For getting integer random values with fairly selection for all nSIV:
ܸܵܫ௦ሺ1: ݇ሻ ൌ ݅݀݊ܽݎሺ݊, 1, ݊ሻ (13)
This MATLAB code given in Eq. 13 is equivalent to the previous code in Eq. 12, but with
significant enhancement. The same analysis is done for this code, and the result is shown in Fig.
4b. It is clearly seen that the selection process for the migrated and mutated nSIV is enhanced.
Note that, this part of modification is not available for 1-dimensional problems, and has less
effect for 2-dimensional problems.
4.2. Clear Duplication Process of the Mutation Stage
According to the original BBO [21], only the worst solutions are mutated. The range of these
mutated solutions can be defined through the preceding mutation algorithm in Section 3.2 as:
For ݉ ൌ ݀݊ݑݎ ቂ݈݁݊݃ݐℎ ቀ
ூ௦ௗ௦
ௗ
ቁ : ݈݁݊݃ݐℎሺݏ݈݀݊ܽݏܫሻቃ
Do mutation (refer to its algorithm)
end for
9. 149 Computer Science & Information Technology (CS & IT)
If d=2, then the worst half of the solutions are to be mutated; and as
the total mutated solutions increase
Figure 4. The Original and Modifi
The modification of this part is to let the clear duplication process be done only on the mutated
solutions, so that the nSIV of the non
this modification, the migrated features
5. PERFORMANCE COMPARISON
The original and modified versions of PMB
of test functions with different dimensions and complexities, where all the details of each test
function are given in the Appendix.
Table 1a shows the parameters used
in [13,16], but with more restriction on the generation limits, which are listed in Table 1b
Computer Science & Information Technology (CS & IT)
, then the worst half of the solutions are to be mutated; and as d increases, the percentage of
mutated solutions increases too.
. The Original and Modified SIV-Selection Process for Migration and Mutation Stages
The modification of this part is to let the clear duplication process be done only on the mutated
of the non-mutated solutions are kept away from any change. Without
features of those non-mutated islands will be corrupted.
OMPARISON
The original and modified versions of PMB-BBO have been tested through a wide variety
of test functions with different dimensions and complexities, where all the details of each test
Appendix.
used for both BBOs. These parameters are similar to those
striction on the generation limits, which are listed in Table 1b
, the percentage of
Migration and Mutation Stages
The modification of this part is to let the clear duplication process be done only on the mutated
mutated solutions are kept away from any change. Without
will be corrupted.
BBO have been tested through a wide variety range
of test functions with different dimensions and complexities, where all the details of each test
are similar to those used
striction on the generation limits, which are listed in Table 1b.
10. Computer Science & Information Technology (CS & IT) 150
Table 1a. BBOs’ parameters (For more details refer to [13,16])
Parameter Value
Population size – or k 50
Max. λ – or I 1
Max. µ – or E 1
mmax 0.01
Elitism 1
Mutation range ݀݊ݑݎ ൬
݇
2
൰ : ݇൨
Number of trails 30
Table 1b. Required generations for various n-dimensional problems
Problem’s dimension # of generations
1, 2, 3, 4, 5 or 6 1,000
8, 9 or 10 5,000
15, 17 or 20 10,000
30 20,000
60 50,000
The performance comparison are evaluated using 120 test functions. They are split into two
equally groups; the first group contains only 2-dimensional test functions and are listed in Table
2a, while the second group contains other n-dimensional test functions and are listed in Table 2b.
The reason for this arrangement is because this paper is a part of a project that deals with 2-
dimensional engineering problem, and the obtained result from this paper will be used a basis to
determine if this essential modification gives a better performance or not before implementing it
for solving that particular problem.
Therefore, more effort was done on 2-dimensional test functions to cover a gradient of difficulties
of unimodal and multimodal functions in conjunction with few and many local minima as traps,
where the diversity of variable bounds through narrow and wide search spaces provides other
challenges.
Table 2a and Table 2b give the best, mean and standard deviation of the 120 test functions. As an
overall, the obtained results shows that the performance of PMB-BBO can be enhanced
effectively by applying the proposed essential modification on its migration and mutation stages
with resizing the range of clear duplication process.
However, for 1-dimensional problems, this proposed modification is not effective because they
have only one SIV, which means that the migration and mutation stages are processed within one
independent variable. Thus, the first part of the proposed modification is absent. Based on this
study, this new version of PMB-BBO can do well if both parts of the modification are embedded.
This can be seen from the first eight test functions shown in Table 2b. Similarly, for other n-
dimensional problems (where ݊ ≥ 2 ), the modified PMB-BBO shows better performance and
wins in most test functions. From the results of the best error, the mean, and the standard
deviation, it can concluded that the proposed correction on the randomized selection process of
the migration and mutation algorithms improves its exploration and exploitation. Fig. 5 shows the
curves of fitness functions of both versions for the Generalized Rastrigin’s function, Schwefel’s
problem 1.2, Qing’s function and Salomon’s function. It can be clearly seen that the modified
PMB-BBO can converge to a better solution more than the original version.
The modified PMB-BBO has proved that this essential modification is highly recommended for
problems with dimension higher than 1. Even, for the few test functions, where the original
version shows better results, the modified version gives a competitive results.
13. 153 Computer Science & Information Technology (CS & IT)
Although, the modified version of
functions, like Price’s Transistor, Storn's
Bent Cigar, Qing, Generalized Rosenbrock and Schwefel F1.2 problems, both
converge to the optimal solutions.
the original version on most of these test functions.
(a)
(c)
Figure 5. Curves of fitness functions of the original and modified PMB for some selected functions
05,
Table 3 shows the CPU time comparison between the modified PMB
BBO. It can be clearly seen from Table 3
the CPU time, which means that
(SPMB-BBO) by around 24.76% [20].
Table 3. Normalized CPU times on 60
5.1. Discussions
As a comparison between the four original models of BBO
PMB-BBO gives the best performance when the given
limits of search spaces, high-dimensional and/or the
[20]. However, PMB-BBO lacks the exp
Original ver.
Hyper-Ellipsoid 1.4617E+00
Qing 1.4605E+00
Salomon 1.5104E+00
Avg. CPU Time 1.4775E+00
Function
Computer Science & Information Technology (CS & IT)
version of PMB-BBO shows enhanced results, for some very hard test
Transistor, Storn's Chebyshev, Trid “or Neumaier F3”, Normalized Rana,
Generalized Rosenbrock and Schwefel F1.2 problems, both version
converge to the optimal solutions. However, as a comparison, the proposed version
these test functions.
(b)
(d)
Curves of fitness functions of the original and modified PMB for some selected functions
05, (b) f30-11, (c) f60-02, (d) f60-03
Table 3 shows the CPU time comparison between the modified PMB-BBO and the original PMB
from Table 3 that the modified version can save around 32.
that it is faster than even the simplified partial migration model
% [20].
Normalized CPU times on 60-dimensional test functions
As a comparison between the four original models of BBO (PMB, SMB, SPMB and SSMB),
BBO gives the best performance when the given problem is hard, has large upper and lower
dimensional and/or the number of islands or population
BBO lacks the exploration [18]. Therefore, in this study, the root problem
Original ver. Modified ver.
1.4617E+00 1.0000E+00 31.58437697
1.4605E+00 1.0000E+00 31.5305639
1.5104E+00 1.0000E+00 33.79377207
1.4775E+00 1.0000E+00 32.3195104
PMB-BBO Versions CPU Time Saving
(%)
some very hard test
Chebyshev, Trid “or Neumaier F3”, Normalized Rana,
versions failed to
version outperform
Curves of fitness functions of the original and modified PMB for some selected functions.(a) f30-
BBO and the original PMB-
that the modified version can save around 32.32% of
it is faster than even the simplified partial migration model
(PMB, SMB, SPMB and SSMB),
is hard, has large upper and lower
number of islands or population size is small
]. Therefore, in this study, the root problem
14. Computer Science & Information Technology (CS & IT) 154
that causes the poor exploration is solved by using an integer random function which provides a
pseudorandom integers from a uniform discrete distribution. Furthermore, the exploitation is
improved by keeping the non-mutated solutions away from any corruption by clear duplication
process.
6. CONCLUSIONS AND SUGGESTIONS
This study proposed some modifications to improve the performance of the original form of
PMB-BBO without using any complicated models for immigration and emigration rates. It is
shown that the simplified linear model still can give good results if the root problems of the
migration and mutation stages are solved. An extensive testing of the original and the proposed
modified versions of PMB-BBO through 120 test functions shows that the performance of the
modified version of PMB-BBO is better on Best, Mean, and StdDev than that of the original
version.
The proposed modification can be used as a basis for modifying the existing modified BBOs in
other literatures. For example, the blended-BBO which is presented in [14] can be a great add-
onto this modified PMB-BBO. Furthermore, even if it is required to employ the complicated
models of immigration and emigration rates, such as the generalized sinusoidal migration modelin
[16].The generalized sinusoidal migration model is based on an old study that was done by James
A. MacMahon in 1987 [19], and was mentioned in ch.3 of [4]. This complicated model shows
great performance, and it is very interesting to re-test this model with considering the proposed
essential modifications that are described in this study.
APPENDIX
This appendix contains a complete list of all the involved test functions that are collected from
various sources where some of them are hard to be found while the other, especially the old
functions, are corrected analytically before being used here. This is because they are available in
their original sources with approximated global solutions. Perhaps due limited computing
capability in that time. These 120 test functions are spread through references [22-64], and some
of these references have a collection of test functions, which means that the popular test functions
can be found in different locations, while the others are located in one or few locations. In
addition, sometimes the information about test functions are available in different references. For
more information regarding any test function, please refer to the related reference(s).
f # Function Name
Dimension
( n )
Variables Bounds Global Optimum References
f1-01 Mineshaft F1 1 0 ≤ ݔ ≤ 10 1.380487165157852 [63]
f1-02 Mineshaft F2 1 −10 ≤ ݔ ≤ 10 −1.416353520337699 [63]
f1-03 Shekel's F1 1 0 ≤ ݔ ≤ 10
−10.1531987550848817763568
39400251
[23,24]
f1-04 Shekel's F2 1
0 ≤ ݔ ≤ 10
−10.4028220447077753290705
18200751
[23,24]
f1-05 Shekel's F3 1
0 ≤ ݔ ≤ 10
−10.5362902992947171054273
57601002
[23,24]
f1-06
Strongin-Zilinskas-
Shaltyanis
1 3 ≤ ݔ ≤ 7.5 −1.601307546494396 [25]
f1-07 Suharev 1 0 ≤ ݔ ≤ 1 −1 [25]
f1-08 ZilinskasF2 1 0 ≤ ݔ ≤ 1 −1.125 [25]
f2-01 Aluffi-Pentini 2 −10 ≤ ݔ ≤ 10 −0.352386073800034 [26]
f2-02 Banana Shape 2 −1.5 ≤ ݔଵ ≤ 1.5
−2.5 ≤ ݔଶ ≤ 0.5
−25 [27]
f2-03 Beale 2 −4.5 ≤ ݔ ≤ 4.5 0 [28,50]
17. 157 Computer Science & Information Technology (CS & IT)
f30-07
GeneralizedSchwefel
F2.26
30
−500 ≤ ݔ ≤ 500
−418.982887272433799807913
601398݊
[22,23,26,28,30,
31,34,36,53,60,62,
65]
f30-08 Mishra F1 30 0 ≤ ݔ ≤ 1 2 [34]
f30-09 Mishra F2 30 0 ≤ ݔ ≤ 1 2 [34]
f30-10
Quartic (or De Jong’s
F4) 30
−1.28 ≤ ݔ ≤ 1.28 0 [23,31,37,46,56]
f30-11 Schwefel F1.2 30 −100 ≤ ݔ ≤ 100 0 [23,31,37,53,62]
f30-12 Schwefel F2.21 30 −100 ≤ ݔ ≤ 100 0 [23,31,37,62]
f30-13
Schwefel F2.22 30
−10 ≤ ݔ ≤ 10 0
[23,30,31,37,53,
62]
f30-14
Sphere(Square Sum,
Harmonic or De Jong’s
F1)
30
−100 ≤ ݔ ≤ 100 0
[23,28,31,36,37,
46,53,57,62]
f30-15
Step 30
−100 ≤ ݔ ≤ 100 0
[23,28,31,37,53,
57]
f60-01 Hyper-Ellipsoid 60 −1 ≤ ݔ ≤ 1 0 [57]
f60-02 Qing 60 −500 ≤ ݔ ≤ 500 0 [53]
f60-03 Salomon 60 −100 ≤ ݔ ≤ 100 0 [26,53]
REFERENCES
[1] Dan. Simon, "Biogeography-based optimization," IEEE Trans. On Evolutionary Computation, vol. 12, no.
6, pp. 702-713, Dec. 2008.
[2] Robert H. MacArthur and Edward O. Wilson, "An Equilibrium Theory of Insular Zoogeography," Journal
of Evolution, vol. 17, no. 4, pp. 373-387, Dec. 1963.
[3] Robert H. MacArthur and Edward O. Wilson, The Theory of Island Biogeography. Princeton, New Jersey:
Princeton University Press, 1967.
[4] Mark V. Lomolino, Brett R. Riddle and James H. Brown, Biogeography, 3rd ed. Sunderland,
Massachusetts: Sinauer Associates Inc., 2009.
[5] R. J. Lincoln, G. A. Boxshall and P. F. Clark, A dictionary of Ecology, Evolution and Systematics.
Cambridge, UK: Cambride University Press, 1982.
[6] PaulamiMaiti and Prabodh K. Maiti, Biodiversity: Perception, Peril and Preservation. Prentice-Hall of
India, 2011.
[7] William A. Nierenberg, Encyclopedia of Environmental Biology, vol. 2. San Diego, California: Academic
Press Inc., 1995.
[8] Jonathan B. Losos and Robert E. Ricklefs, The Theory of Island Biogeography Revisited. Princeton, New
Jersey: Princeton University Press, 2010.
[9] Alan A. Myers and Paul S. Giller, Analytical Biogeography: An Integrated Approach to the Study of
Animal and Plant Distributions. London, UK: Chapman and Hall, 1990.
[10] Martin L. Cody, Plants on Islands: Diversity and Dynamics on a Continental Archipelago. Berkeley and
Los Angeles, California: University of California Press, 2006.
[11] Robert H. MacArthur and Joseph H. Connell, The Biogeography of Populations. New York: John Wiley &
Sons Inc., 1966.
[12] Robert H. MacArthur, Geographical Ecology: Patterns in the Distribution of Species. New York: Harper &
Row Publishers Inc., 1972.
[13] Haiping Ma, "An analysis of the equilibrium of migration models for biogeography-based optimization,"
Information Sciences, vol. 180, no. 18, pp. 3444-3464, 15 Sept. 2010.
[14] Haiping Ma and Dan. Simon, "Blended biogeography-based optimization for constrained optimization,"
Engineering Applications of Artificial Intelligence, vol. 24, no. 3, pp. 517-525, Apr. 2011.
[15] Dan. Simon, "A probabilistic analysis of a simplified biogeography-based optimization algorithm,"
Evolutionary Computation, vol. 19, no. 2, pp. 167-188, Summer 2011.
[16] Haiping Ma and Dan. Simon, "Analysis of migration models of biogeography-based optimization using
Markov theory," Engineering Applications of Artificial Intelligence, vol. 24, no. 6, pp. 1052-1060, Sept.
2011.
[17] Dan. Simon, M. Ergezer and Dawei Du, "Population distributions in biogeography-based optimization
algorithms with elitism," IEEE International Conference on Systems, Man and Cybernetics, pp. 991-996,
Oct. 2009.
18. Computer Science & Information Technology (CS & IT) 158
[18] Wenying Gong, ZhihuaCai and Charles X. Ling, "DE/BBO: A hybrid differential evolution with
biogeography-based optimization for global numerical optimization," Soft Computing, vol. 15, no. 4, pp.
645-665, Apr. 2011.
[19] James A. McMahon, "Disturbed lands and ecological theory: an essay about a mutualistic association," in:
Restoration Ecology, a Synthetic Approach to Ecological Research, William R. Jordan III, Michael E.
Gilpin and John D. Aber. Cambridge, UK: Cambride University Press, 1987, pp. 221–237.
[20] Ali R. Alroomi, Fadhel A. Albasri and Jawad H. Talaq, “Performance Comparison between the Original
Forms of Biogeography-Based Optimization,” to be published, Sept. 2013.
[21] Dan. Simon, "The Matlab Code of Biogeography-Based Optimization," [online] Aug. 2008,
http://academic.csuohio.edu/simond/bbo/(Accessed: 01 Feb. 2013).
[22] Wen Wan and Jeffrey B. Birch, "Using a modified genetic algorithm to find feasible regions of a
desirability function," Quality and Reliability Engineering International, vol. 27, no. 8, pp. 1173-1182, Dec.
2011.
[23] Xin Yao, Yong Liu and Guangming Lin, "Evolutionary Programming Made faster," IEEE Trans. on
Evolutionary Computation, vol. 3, no. 2, pp. 82-102, Jul. 1999.
[24] Chien-Wen Chao, Shu-CherngFand and Ching-Jong Liao, "A Tropical cyclone-based method for global
optimization," Journal of Industrial and Management Optimization, Vol. 8, no. 1, Feb. 2012.
[25] Elena Kiseleva and Tatyana Stepanchuk, “On the Efficiency of a Global Non-differentiable Optimization
Algorithm Based on the Method of Optimal Set Partitioning,” Journal of Global Optimization, vol. 25, no.
2, pp. 209–235, Feb. 2003.
[26] M. Montaz Ali, CharoenchaiKhompatraporn and Zelda B. Zabinsky, "A Numerical Evaluation of Several
Stochastic Algorithms on Selected Continuous Global Optimization Test Problems" Journal of Global
Optimization, vol. 31, no. 4, pp. 635-672, Apr. 2005.
[27] Ray-Bing Chen, Weichung Wang and Fuhuang Tsai, "A Basis-based Response Surface Method for
Computer Experiment Optimization," 2006
[28] Milos Subotic, Milan Tuba and NadezdaStanarevic, "Different approaches in parallelization of the artificial
bee colony algorithm," International Journal of Mathematical and Methods in Applied Sciences, vol. 5, no.
4, Mar. 2011.
[29] Celso A. G. Santos, Paula K. M. M. Freire and Sudhanshu K. Mishra, "Cuckoo search via Le ́vy Flights for
optimization of a physically-based runoff-erosion model," Journal of Urban and Environmental
Engineering, vol. 6, no. 2 , pp. 123-131, Dec. 2012
[30] Xinchao Zhao and Xiao-Shan Gao, "Affinity genetic algorithm," Journal of Heuristics, vol. 13, no. 2, pp.
133-150, Apr. 2007.
[31] Ernesto P. Adorio. (2005, Jan.), MVF - Multivariate Test Functions Library in C for Unconstrained Global
Optimization. University of the Philippines Diliman. Quezon City, Metro Manila, Philippines. Available:
http://geocities.ws/eadorio/mvf.pdf
[32] RachidChelouah and Patrick Siarry, "Tabu Search applied to global optimization," European Journal of
Operational Research, vol. 123, no. 2, pp. 256–270, Jun. 2000.
[33] Sudhanshu Mishra. (2006, Aug.). Some new test functions for global optimization and performance of
repulsive particle swarm method. North-Eastern Hill University. Shillong, India. Available:
http://mpra.ub.uni-muenchen.de/2718/1/MPRA_paper_2718.pdf
[34] Andrea Gavana, "Test Functions Index," [online] Feb. 2013, http://infinity77.net/
global_optimization/test_functions.html (Accessed: 01 April 2013).
[35] Z. Emami, M. H. Farahi and E. Ansari. (2012, Feb.). Global Optimization with Electro Weak. The Open
Industrial and Manufacturing Engineering Journal. [Online]. vol. 5, pp. 1-7. Available:
http://benthamscience.com/open/toimej/articles/V005/1TOIMEJ.pdf
[36] MarcinMolga and CzesławSmutnicki. (2005, Apr.). Test functions for optimization needs. North-Eastern
Hill University. Shillong, India. Available: http://www.zsd.ict.pwr.wroc.pl/files/ docs/functions.pdf
[37] ShahryarRahnamayan, Hamid R. Tizhoosh and Magdy M. A. Salama, "A novel population initialization
method for accelerating evolutionary algorithms," Computers & Mathematics with Applications, vol. 53,
no. 10, pp. 1605–1614, May 2007
[38] Leo Breiman and Adele Cutler, "A deterministic algorithm for global optimization," Mathematical
Programming, vol. 58, no.1-3, pp. 179-199, Jan. 1993.
[39] Mohamed B. Trabia and Xiao Bin Lu, "A Fuzzy Adaptive Simplex Search Optimization Algorithm,"
Journal of Mechanical Design, vol. 123, no. 2, pp. 1-10, Jun. 2001.
[40] J. B. Lee and B. C. Lee, "A global optimization algorithm based on the new filled function method and the
genetic algorithm," Engineering Optimization, vol. 27, no. 1, pp. 1-20, 1996.
[41] M. Fernanda P. Costa, Florbela P. Fernandes, Edite Manuela da G. P. Fernandes, "A deterministic-
stochastic method for nonconvex MINLP problems," 2nd International Conference on Engineering
Optimization, Sept. 6-9, 2010, Lisbon, Portugal.
19. 159 Computer Science & Information Technology (CS & IT)
[42] Kaj Madsen, "Test problems for global optimization," [online] Jun. 2008, http://www2.imm.
dtu.dk/~kajm/Test_ex_forms/test_ex.html (Accessed: 09 April 2013).
[43] M. N. Vrahatis, D. G. Sotiropoulos and E. C. Triantafyllou, "Global optimization for imprecise problems,"
in: Developments in Global Optimization, Nonconvex Optimization and Its Applications, Immanuel M.
Bomze, TiborCsendes, Reiner Horst, Panos M. Pardalos. Springer Science+Business Media Dordrecht:
Kluwer Acdemic Publishers, 1997, vol. 18, pp. 37-54.
[44] Baker Kearfott, "An efficient degree-computation method for a generalized method of bisection,"
NumerischeMathematik, vol. 32, no. 2, pp. 109-127, 1979.
[45] EsinOnbaşoğlu and LinetÖzdamar, "Parallel Simulated Annealing Algorithms in Global Optimization,"
Journal of Global Optimization, vol. 19, no. 1, pp. 27-50, Jan. 2001.
[46] S. Rahnamayan, H. R. Tizhoosh and M. M. A. Salama, "Opposition-Based Differential Evolution for
Optimization of Noisy Problems," IEEE Congress on Evolutionary Computation, CEC 2006, pp.1865-
1872, Jul. 2006.
[47] Klaus Müller and Leo D. Brown, "Location of saddle points and minimum energy paths by a constrained
simplex optimization procedure," Theoreticachimicaacta, vol. 53, no. 1, pp. 75-93, 1979.
[48] Wan-Ching Chou, "Multiple Optima Problems in Higher Dimensions," [online] Mar.
2013,http://www.math.ntu.edu.tw/~wwang/cola_lab/test_problems/multiple_opt/multiopt_prob/ (Acc-
essed: 25 April 2013).
[49] MathWorks, "Global Optimization Toolbox," [online] 2013, http://www.mathworks.com/ products/global-
optimization/index.html (Accessed: 25 April 2013).
[50] Singiresu S. Rao, Engineering Optimization, Theory and Practice, 4th ed. Hoboken, New Jersey: John
Wiley & Sons Inc., 2009.
[51] Gisela C.V. Ramadas and Edite M.G.P. Fernandes, "Solving Nonlinear Equations by a Tabu Search
Strategy," 11th International Conference on Computational and Mathematical Methods in Science and
Engineering, CMMSE2011, vol. 4, no. 1, pp. 1578-1589, Jun. 2011.
[52] Jorge J. Moré, Burton S. Garbow and Kenneth E. Hillstrom, "Testing Unconstrained Optimization," ACM
Transactions on Mathematical Software (TOMS), vol. 7, no. 1, pp. 17-41, Mar. 1981.
[53] Anyong Qing, Differential Evolution, fundamentals and Applications in Electrical Engineering. Singapore:
Wiley-IEEE Press, Sept. 2009.
[54] Muhammad Aria, "Educational Simulator for Teaching of Particle Swarm Optimization in LabVIEW,"
TELEKONTRAN, vol. 1, no. 1, Jan. 2013.
[55] Sudhanshu Mishra. (2006, Oct.). Repulsive Particle Swarm Method on Some Difficult Test Problems of
Global Optimization. North-Eastern Hill University. Shillong, India. Available: http://mpra.ub.uni-
muenchen.de/1742/1/MPRA_paper_1742.pdf
[56] V. Zitko. (1984, Sept.), “Simplex Optimization,” Canadian Technical Report of Fisheries and Aquatic
Sciences, No. 1308. Fisheries and Oceans. Biological Station (St. Andrews, N.B.), Canada. Available:
http://www.dfo-mpo.gc.ca/Library/20701.pdf
[57] Rainer Storn and Kenneth Price, "Differential Evolution – A Simple and Efficient Heuristic for Global
Optimization over Continuous Spaces," Journal of Global Optimization, vol. 11, no. 4, pp. 341-359, Dec.
1997.
[58] Ashok D. Belegundu and Tirupathi R. Chandrupatla, Optimization Concepts and Applications in
Engineering, 2nd ed. New York: Cambridge University Press, 2011. Page331
[59] Kenneth V. Price, Rainer M. Storn and Jouni A. Lampinen, Differential Evolution: A Practical Approach to
Global Optimization, Berlin, Germany: Springer Science+Business Media, 2005.
[60] Sudhanshu Mishra. (2012, Aug.). Global optimization of some difficult benchmark functions by cuckoo-
host co-evolution meta-heuristics. North-Eastern Hill University. Shillong, India. Available:
http://mpra.ub.uni-muenchen.de/40666/1/MPRA_paper_40666.pdf
[61] Darrell Whitley, Deon Garrett and Jean-Paul Watson, "Quad Search and Hybrid Genetic Algorithms," in:
Genetic and Evolutionary Computation--GECCO 2003, G. Goos, J. Hartmanis and J. Van Leeuwen. Berlin,
Germany: Springer Science+Business Media, 2003, vol. 2, pp. 1469-1480.
[62] A. K. Qin, V. L. Huang and P. N. Suganthan, "Differential Evolution Algorithm with Strategy Adaptation
for Global Numerical Optimization," IEEE Trans. On Evolutionary Computation, vol. 13, no. 2, pp. 398 -
417, Apr. 2009.
[63] David R. Monismith Jr., “The Uses of the Slime Mold Lifecycle as a Model for Numerical Optimization,”
Ph.D. dissertation, Dept. Elect. Eng., Oklahoma State University, Oklahoma City, OK, 2010.
[64] DebasisKundu and Amit Mitra, "Estimating the parameters of the linear compartment model," Journal of
Statistical Planning and Inference, vol. 70, no. 2, pp. 317-334, May 1998.
[65] Lappeenranta University of Technology, "The function testbed," [online] May. 2007,
http://www.it.lut.fi/ip/evo/functions/functions.html (Accessed: 10 April 2013).
20. Computer Science & Information Technology (CS & IT) 160
Authors
Ali Ridha Alroomi received the B.Sc. degree in Process Instrumentation and Control
Engineering from University of Bahrain, Bahrain, in 2006. In the period between 2009-
2013, he covered the electrical courses of the B.Sc. program in Electrical Engineering,
and followed by M.Sc. degree from the same university. After B.Sc. graduation, he was
selected to be as a project engineer in Moore Control & Engineering “MCE Middle East”
till the mid of 2007, when he joined Yokogawa Middle East as a DCS subsystem and
graphics engineer. From 2008 to 2012, he worked in Aluminum Bahrain (ALBA B.SC.)
as an instrumentation and control engineer in its power plants, generation and auxiliary C&I maintenance.
He is currently working as a researcher in the field of electrical power systems and evolutionary
computation.
Fadhel Abbas Al-Basri received the B.Sc. and M.Sc. degrees in Electrical
Engineeringfrom University ofBahrain, Bahrain, and Ph.D. degree in the same field from
University of Western Ontario, Canada,in 1992, 1997 and 2007, respectively. He worked
in Ministry of Electricity and Water, Bahrain, as an electrical engineer from 1993 to 1994.
In 1994, he joined the University ofBahrain as teaching and research assistant and
currently an assistant professor in the department of electrical engineering. His research
interest is power systems protection, power systems analysis andFACTS-devices.
Jawad Hasan Talaq received B.Sc. degree from University of Technology, Baghdad, Iraq
(1981), M.Sc. degree from University of Strathclyde,Glasgow, UK (1987), and Ph.D.
degree from Technical University of Nova Scotia, Halifax, Canada (1993); all in
electrical engineering. He worked in Ministry of Electricity and Water, Bahrain, as a
graduate engineer in Riffa power plant from 1981 to 1985, and as a shift charge engineer
from 1985 to 1989, when he moved from industrial field to academic field. He is now an
associate professor in the department of electrical engineering, University of Bahrain. His
teaching covers power systems operation and control, power systems dynamics, and advanced power
systems analysis; and his research topics include power systems operation, control and dynamics, and
applied soft computing.
![Sundarapandian et al. (Eds) : ICAITA, SAI, SEAS, CDKP, CMCA-2013
pp. 141–160, 2013. © CS & IT-CSCP 2013 DOI : 10.5121/csit.2013.3812
ESSENTIAL MODIFICATIONS ON
BIOGEOGRAPHY-BASED OPTIMIZATION
ALGORITHM
Ali R. Alroomi1
, Fadhel A. Albasri2
and Jawad H. Talaq3
1
Electrical Engineer, University of Bahrain, Sakhir, Bahrain
wadyan@gmail.com
2
Department of Electrical Engineering, University of Bahrain, Sakhir, Bahrain
falbasri@uob.edu.bh
3
Department of Electrical Engineering, University of Bahrain, Sakhir, Bahrain
jhstallaq@eng.uob.bh
ABSTRACT
Biogeography-based optimization (BBO) is a new population-based evolutionary algorithm and
is based on an old theory of island biogeography that explains the geographical distribution of
biological organisms. BBO was introduced in 2008 and then a lot of modifications were
employed to enhance its performance. This paper proposes two modifications; firstly,
modifying the probabilistic selection process of the migration and mutation stages to give a
fairly randomized selection for all the features of the islands. Secondly, the clear duplication
process after the mutation stage is sized to avoid any corruption on the suitability index
variables. The obtained results through wide variety range of test functions with different
dimensions and complexities proved that the BBO performance can be enhanced effectively
without using any complicated form of the immigration and emigration rates. This essential
modification has to be considered as an initial step for any other modification.
KEYWORDS
Biogeography-Based Optimization, BBO, Evolutionary Algorithm, Migration, Partial Migration
1. INTRODUCTION
The science of biology becomes one of the main resources of inspiration to develop the modern
optimization techniques, such as ant colony optimization (ACO), bee colony optimization (BCO),
wasp swarm optimization (WSO), bacterial foraging optimization (BFO), genetic algorithm (GA),
evolutionary strategy (ES), differential evolution (DE), particle swarm optimization (PSO), etc.
Biogeography-based optimization (BBO) is a new population-based evolutionary algorithm (EA)
that was introduced by Dan Simon in 2008 [1].
The algorithm of BBO is based on an old theory of island biogeography that explains the
geographical distribution of biological organisms. This theory was established by H. MacArthur
and Edward O. Wilson during their exploration study in the period between 1960-1967 [2,3].](https://image.slidesharecdn.com/csit3812-180129043632/85/ESSENTIAL-MODIFICATIONS-ON-BIOGEOGRAPHY-BASED-OPTIMIZATION-ALGORITHM-1-320.jpg)
![Computer Science & Information Technology (CS & IT) 142
Island, in biogeography, is any area of suitable habitat (local environment occupied by an
organism [5]) surrounded by an expense of unsuitable habitat and is endowed with exceptionally
rich reservoirs of endemic, exclusive, strange and relict species [6].
Each island has its own features as simple biotas, varying combinations of biotic and abiotic
factors, and variability in isolation, shape, and size [7,9].
In BBO, the islands (or habitats) are the dependent variables, and the features availability on
those islands represents the independent variables.
The original BBO has a weakness in its migration and mutation stages that affects its
performance. The objective of this paper is to solve that weakness by focusing on the root
problems instead of using other complicated approaches, such as the complicated migration
models represented in [13,16].
This paper is organized as follows: Section 2 gives a brief overview about the theory of island
biogeography and Section 3 explains the algorithm of BBO. Section 4 shows how BBO
performance can be enhanced more by solving the root problems of the migration and mutation
stages, and then followed by a performance comparison between the original and the modified
BBO in Section 5. The conclusions and suggestions are presented inSection 6.
2. THE THEORY OF ISLAND BIOGEOGRAPHY
The equilibrium theory of island biogeography proposes that the number of inhabited species on
an island is based on the dynamic equilibrium between new immigrated species onto an island
and the extinct species out from that island [2,3,8].
Fig. 1 graphically represents the equilibrium model with linear immigration (or speciation) rate λ
and emigration (or extinction) rate µ, where they can be plotted as logistic, exponential or any
proper function [4,10,11].
Figure 1. Simplified equilibrium model of a biota of a single island
I and E are the maximum possible immigration and emigration rates, respectively. Ioccurs when
the island is empty of any species and thus it offers a maximum opportunity to the species on the
other islands for immigrating to settle on it; whereas the arrivals on that island increases, the](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)